L. E. Barrett

P. E. Allaire

A Finite Length Bearing Correction Factor for Short Bearing Theory

E. J. Gunter

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A rapid method for calculating the general nonlinear response of finite-length plain jour­nal and squeeze film damper bearings is presented. The method incorporates a finite­length correction factor which modifies the nonlinear forces obtained from short bearing theory. The steady-state rotational, precessive squeeze, and radial squeeze forces ob­tained with the correction factor compore extremely well with the forces obtained from an analytic solution of Reynolds equation using a variational approach up to LID of 1.25 and hence couer, the most commq,:ily encountered LID ratios. The method is no more time consuming than the 1hort bearing analysis and is especially auited to nonlinear transient analysis of flexible rotors.

l Introduction

The analysis of present-day rotor-bearing systems often requiresthe calculation of nonlinear hydrodynamic bearing and seal forces to insure the adequacy of these components in realizing the objective of acceptable machine performance. Very often the nonlinear analysis must be performed in the time domain using time-transient tech­niques. For example, the determination of the attitude angle of floating ring-type oil seals to obtain the correct force directions in turbo-compressors may require such an analysis (1).

Present interest in squeeze film damper bearings often requires nonlinear bearing analysis. This is primarily due to the fundamental nonlinear nature of squeeze film dampers which, unlike journal bearings, require dynamic journal motion to produce hydrodynamic forces. Thus any linearization which may be done is based on an as­sumption of the dynamical motion. In m0&t instances, circular syn­chronous precession of the damper journal about the bearing center is assumed, but the actual motion may be quite different [2) thus re­quiring time-transient analysis to verify the design parameters.

In most instances, when transient nonlinear hydrodynamic journal bearing analysis is performed, the short bearing form of the Reynolds equation is used because of the time savings over the solution of the complete two-dimensional Reynolds equation. This practice is often justified because the LID ratio of the bearing or seal being analyzed is less than 0.25. For LID ratios greater than this, the error introduced by use of the short bearing IIS8Umption is often tolerated rather than incur the computational expense of solving the two dimensional Reynolds equation with the usual numerical techniques, i.e., finite­difference and fmite-element met.hods (3, 4). Analytic solutions to the two-dimensional Reynolds equation using a variational approach with a Fourier-series expansion of the pressure have been made (5-7). Al·

Contributed by the Lubrication Diviaion of THE AMERICAN SOclETY OF

MECHANICAL ENGINEERS for preaentation at the ASME-ASLE Lubrication Conference, Dayton, Obio, October 16-18, 1979. Reviaed manuscript received by the Lubrication Division, July 5, 1979. Paper No. 79-Lub-13.

Copies will be available until July, 1980.

though computationally faster than the finite-difference and finite­element methods, they are still computationally expensive for non­linear transient analyses of geometrically complex bearings. Imped­ance vectors (8), related to mobility vectors [9), have recently been employed to analyze finite-length plain journal and squeeze film bearings. Based on execution time data in reference (8), the impedance method is about six times faster than numerical integration of the short bearing form of the Reynolds equation used in references (10, 11 ). Approximate solutions to the two-dimensional Reynolds equation have also been obtained by applying an end-leakage correction factor to the infinitely long bearing solution of Reynolds equation (12).

This present paper presents an efficient method for calculating the hydrodynamic forces for finite-length plain journal bearings, seals, and squeeze film dampers. The method consists of applying a finite­length correction factor to the pressure obtained from the short bearing form of the Reynolds equation [ 13). The form of the correction factor is found from the two-dimensional Reynolds equation by as­swning a separable solution of the two-dimensional pressure field. For small LID ratios, the corrected solution reduces to the short bearing solution and justifies the assumption of a separable solution. The basic form of the pressure correction factor in reference (13) has been modified (14) and the method is herein extended to LID ratios up to 1.25. The hydrodynamic forces found using the finite-length corrected short bearing solution are compared to those calculated using the variational analytic solution (5, 61 and the comparison is found to be quite good. If the finite-length correction factor is used in conjunction with the analytic short bearing solutions, the computer execution times for journal transient motion are approximately 2 to 3 times less than those using the impedance method. The present method ia ideally suited for transient analyses of rotor-bearing sys­tems over a very useful range of bearing LID ratios.

2 Analysis

%.1 Correction Factor. Subject to tbe normal assumptions of laminar, incompressible isoviscous flow, Reynolds equation for an

1

I Nm,Pmh,.,. Q 1 Q70

Dyrobes Rotordynamics Software http://dyrobes.com

y

x

JOU RNA L y BEARI N G

.......

0

L - ---i

Fig. 1 Schematic of plaln journal bearing

axially aligned finite-length journal bearing shown in Fig. 1 is

6µ8-- - h3- -h3-= 0 1 () [ ()pl '()2p R 2 ~ C>O C>z 2

(1)

where

• 'C>h C>h B • (w-2tf>) - + 2-

'C>O 'C>t

Denoting the pressure for instantaneous values of h and '()h/'()t by (13]

p(O, z) = f(z)p. (8) (2)

where pc(O) is the circumferential pressure around the axial centerline, equation (1) can be written

where

b+ - f- - = O (ID2 d2f

R dz 2

b = 6µ8 p,h3

c2 .. -=..!...~ lh3 "P·] p.h3 ~ ClO

The boundary conditions for equation (3) are

f = Oatz = -L/2

f = 0 at z .. +L/2

(3)

and the solution to equation (3) becomes

I. -b (!!.)2 (i _ cosh (Gz/R) J G cosh (GL/D)

(4)

T he preasure in the bearing is therefore

(8 z) • -6µB (!!.)2 [i _ cosh(Gz/R)] P ' hs G cosh(GL/ D)

(5)

For small LID (and hence z/D), the term

(!!_)2 [l _ cosh (Gz/R)J G cosh (GL/D)

approaches L2/ 4 - z2, and equation (5) reduces to

p(8, z) = -!~B 1~2

- z2] (6)

which is t he pressure obtained from the short bearing form of Reyn­olds equation (15).

The te rm

(!!.)2 [l _ cosh (Gz/R >j G cosh (GL/D)

in equation (5) has the function of modifying the centerline pressure obtained from the short be.ar ing solution to Reynolds equation for finite LID, and the remainder of the equation is proportional to the short bearing centerline pressure, i.e.,

µB -4 h 3 = 3L2 Pot (7)

Equation (5) can therefore be written

( D )21 cosh (GzL/D)l

p(O, z) • 2Poc CL l - cosh (GL/D) (8)

where the substitution, 'i = 2z/L, has been made. Equation (8) suggests a form of a correction factor which may be used to modify the pressure obtained from t he short bearing solution of Reynolds equation for finite length bearings.

2.2 Steady-Stat e Analysis. A solution to equation (8) can be obtained once the functional form of G is known. Observing that as LID - oo, df /dz -+ 0, f .... l and from equation (3), G2 becomes

-6µ8R 2 0 2 • g2 = LID » l (9)

p.h3 .

where p. is the pressure obtained from the infinitely long approxi­mation of Reynolds equation. Under steady-state operation,

C>h B=w-

'C>8

(R)2 wl(2 + ( cos 8) sin 8

p . • 6µ -c (2 + f 2)(1+lcos0)2

and equation (9) becomes

2 (2 + , 2)

g .. ( l + ( cos 0)(2 + ( cos 8)

(10)

---------Nomenclature--------------------------------------------------------------------------b = 6µB/p.h3, L-2 B = (w - 24>)C>h/Cl8 + 2C>h/C>t, L T- 1

c = bearing radial clearance, L D = bearing diameter, L f = axial pressure function, d im F, = radial bearing force, F F1 = t.angential bearing force, F g, G = defined by equation (3) and equat ion

(10), dim g,, g, = defined by equations (11). (12). (23),

(24), dim

2

h .. bearing film thickness, L L = bearing length, L p • bearing pressure, FL - 2

Poe • short bearing center-line pressure, FL-2

p, '"' rad ial pressure component, FL - 2

p, = tangential pressure component, FL - 2

R • bearing radius, L S .. Sommerfeld number, dim S. = 4S(L/D)2, dim W .. journal weight, F

z "" axial coordinate, L i = 22/L, dim 'Y•.2 =angles defined by equations (23), (24),

dim ' = eccentricity ratio, dim <-= dddt, rr- • 0 .. circumferential coordinate, dim µ .. dynamic viscosity, FTL - 2

1/- ""journal attitude angle, dim w =journal angular velocity, T- 1

~- .rna l precession rate, T - 1

Transactions of the ASME

Noting that for small LID equation (8) reduces to equation (6) in­dependent of the functional form of 1. and that equation (10) is valid for large LID, it is assumed here that equation (10) is reasonably valid for intermediate values of LID. Equation (10) indicates that 1 is a function of both E and 8. In the present form, direct substitution of I into equation (8) would present severe problems in obtaining an analytic expression for the hydrodynamic forces by integrating the pressure on the journal surface. To allow for a variation in the steady state journal attitude angle, the following radial and tangential components of I are defined

lr = g cos8

lt ""g sin8

To eliminate explicit dependence of Ir and g1 on 8, "average" values could be defined. However, cloeed form integration of Cr and Ct would be difficult (if not impossible) to perform. Instead, weighted average values, g,2 and It 2, are defined which permit relatively easy closed form integration. They are

1 J:. lr2 = - w(g cos 8)2d8 1r 0

(11)

1 J:. lt2 = - w(g sin 8) 2d8 1r 0

(12)

A weighting function, w = ir, has been found to give best results. The averaging is performed over the range 0 5 8 5 ir since during

steady-state operation the pressure field is antisymmetric about the journal-bearing line of centers using the half-Sommerfeld cavitation condition. In terms of the components of g2 defined by equations (11) and (12), the radial and tangential components of the pressure can be written

_ 2poc cos 8 [l _ cosh (grILID)] Pr - g,2(LID)2 cosh (grLID)

(13)

Pt"' 1---~--2poc sin 0 [ cosh (gtILID)I g, 2(LfD)2 cosh (gtLID)

(14)

The evaluation of g,2and11 2 results in

2 ir(2+t2)1 l 4 l

~ = l+ -f2 (1 - f2)1/2 (4 - f2)1/2

(15)

1 2 _ ir(2 + t 2) [ l _ l I

I - 2 (1 - f2)1/2 (4 - f2)1f2 (16)

The radial and tangential hydrodynamic force components are found by integrating the pressure components over the positive pressure region, i.e.,

LR f I f• F, "'2 _

1 J o p,d8dz (17)

LR f 1 J:r Ft,., _ Ptd8dZ 2 -I 0

(18)

which results in

F, = - 3µwLD3,2 [1 - (_D ) tanh (g,LID)] (19) 2c21,20 - ,2)2 \g,L

Ft = 3µwLD3"t [1 - (_D) tanh (g,LID)l (20)

8c21t2(1 _ ,2)3/2 \gtL

The steady-state operating eccentricity ratio and attitude angle become

S • - 1 - - tanh L D 1 {[ ,2 2 [ ~D ) 12 24ir 21, 2(1 - , 2)2 rL (gr I )

[ irf 2 [ ~D ) 121-112 +

2 2 312 I - - tanh (gtLID)

811 (1 - f ) tL

[

l - f..:_D_) tanh (g,LID)l tan ir = [~)2"(1 ~ft2)19 ___;;\gD_tL -

1 - ~,J tanh (g,L/D)

Journal of Lubrication Technology

(21)

(22)

where the Sommerfeld number, S, ia defined as

S • µwLD (!i)2 21fW c

2.3 Radial Journal Motion. The preceding section describes the application of the finite-length correction factor to steady·atate motion. It is also applicable to steady or instantaneous values pre· cession of the journal (nonzero ~) without modification. Nonzero radial journal motion can be handled in a completely analogous manner, although the functions Ir 2 and It 2 are different from those used for zero radial motion. By using the infinitely long bearing pressure for radial motion in equation (9) and integrating over the positive pressure region, the functions Ir 2 and It 2 become

where

2 2 [ 2<ir--.,.> 8<ir--.,i> Ir = - ir + ---'-'--f 2 (1 - ,2p12 4 - f2)1/2

Ct 2 = 2 ( (1f - "(1) (ir - "(2) ]

(1 - f 2) 1/2 (4 - f2)1/2

-Yt =tan-• [(l -'f2)1~

)'2 = tan-1 [(4 -"2)19

(23)

(24)

2.4 General Dynamic Motion. To calculate the general dy. namic motion of a journal within its bearing, the forces due to in­stantaneous values of journal position and velocity are obtained by superposing the pressures due to rotation-precession and radial squeeze motion and integrating over the resulting net positive pres· sure region. The instantaneous forces are then used in an appropriate dynamical integration routine to predict a subsequent position and velocity. The instantaneous forces are

F, = D3 f~ {1 - f _!!_) tanh i.Jlr1LID)} f," Poct cos 8d0

L jg,, \g,,L 1,

+ ~ {1 - {_!!_) tanh (g,2L/D)} J," P<><2 cos 8d01 (25) lr2 \cr2l 11

Ft "' D3

f~ {1 - f _!!_) tanh (g, 1L/Dl} f," Poet sin 8d8 L lc11 \c11L ••

+ ~ {1 - f _!!_) tanh l.s12LID)} J," p.,.2 sin 8d8] (26) lt2 \gt2L ''

where Poet and P<><2 are the center-line pressures obtained from short bearing theory for rotation-precession and radial squeeze motion respectively and where g,1, g, 1, g,7, and Kt2 are defined by equations (15), (16), (23), and (24), respectively. The angles 81 and 02 define the region of positive pressure and are dependent on the journal motion and cavitation assumptions.

3 Results To evaluate the accuracy of the finite-length correction factor, a

comparison of the steady-state and radial squeeze forces obtained using the correction factor was made with the forces obtained from an analytic solution of finite-length journal bearings (5, 6). Figs. 2 through 4 show the results of the steady-state force comparison for LID ratios of0.5, 1.0, and 1.25. The steady-state operating eccentricity is plotted as a function of the short bearing Sommerfeld number, S,. It is observed that the results with the finite·length correction factor provide a very close approximation to the analytic results even at LID = 1.25. The correlation is best in the middle range of eccentricity ratios (0.2 ~ t ~ 0.8) and hence covers the range most useful in tran· sient analyses. It is also noted that even for LID "" 0.5 the uncorrected short bearing results are significant.ly in error, particularly at high eccentricity ratios. A comparison of attitude angle is shown in Fig. 5.

Figs. 6 through 8 compare the radial forces obtained for pure radial squeeze motion using corrected sh 1 bearing theory with those ob·

3

4

.. -'lu -

L/O• 0.5 - · -FINITE LENGTH

CORRECTION

-- FINITC LENGTH ANAl.YTIC SOl.UTION

Rel (5,6)

- ---- SHORT BEARING THEORY

l .O

Fig. 2 Comparison of 1teady state lolld c:apac:ltle1 lor U O = 0.5

10.0 L/0 • 1.0

-·-FINITE LENGTH CORRECTION

-- FINITE l.CNOTH ANALYTIC SOLUTION

Rel (5,6)

1.0 ----- SHORT BEARING THEORY

~

:'I~

~~ 0.1

001~---~-__. _ _.__.__.._._.__, 01 lO

Fig. 3 Comparison of steady state load capacltlH lor UO = 1.0

90 LIO• l.25

80 - · -FINITE LENGTH

ci CORRECTION

..... 70

- - FINITE LENGTH 0 ANALYTIC SOLUTION

Roi (5,6)

?-60 ---- -SHORT BEARING THEORY

..... !50 _J

C> z 40 ~

..... 30 0

J ... 20 ... ...

~ 10

0 0.1 1-E 1.0

Fig. 4 Comparison ol steady stale load capac:lllH lor UO = 1.25

L/O• 1.25 - · -FINITE l.CNGTH

CORRECTION

-- FINITE LCNOTH ANALYTIC SOl.UTION

Rel (5,6) -----SHORT 8EARING

THEORY

,-l ~ 01 ~ 0 .1

0.01.__ ___ ..__ _ _.__..___.__._...._ ....... ~ 0 .1 I- E 1.0

Fig. 5 Comparison of steady state attitude angles for UO = 1.25

L/0 • 0.5 - · - FINITE l.CNGTH

CORRECTION

-- FINITE l.CNOTH ANALYTIC $01.UTION

Roi (5,6) - - - -- SHORT BEARING

THEORY

Fig. I Comparison of radial squHze lore•• lor U O = 0.5

' ' ' ' ' ' ' ' ' ' '

L/O • 1.0 - ·-FINI TE LENGTH

CORRECTION

-- FINITE LENGTH ANAl.YTIC $01.UTION

Roi (5, 6) ----- SHORT BEARING

THEORY

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' . ' ~- ',

1~---~--...__.__~~~~~ 0.1 1.0

1- E

Fig. 7 Comparison ol radial squeeze lorcH IOI UO = 1.0

Transactions of the ASME

L/0• 1.25 -·-FINITE lENGTH

' CORRECTION

' ... ,, -- FINITE LENGTH ', AHAlYTIC SOlUTIOH

, Roi (5,1) ', -----SHORT BEARING

' THEORY

' ' ', ' ' ' ' ' ' ' ',

' ' ' ' ' ' ', ',

l'--~~~-'-~---''---'-~..._.._.._ ...... ~ 0 .1 1- € 1.0

Fig. 8 Comparison of radial squeeze forces for UD = 1.25

tained analytically for finite length bearings and by uncorrected short bearing theory. Again, the finite -length correction factor compares very favorably with the analytic solution over a wide range of eccen­tricity and LID ratios. For LID ~ 0.25, the results for both steady­state and radial squeeze motion using the three methods are nearly identical, as is expected, and are not shown here.

The finite length correction factor described herein does not result in any appreciable increase in the computational time required for short bearing analysis. The effect of increased LID ratios in plain journal bearings, seals, and squeeze film damper bearings can, therefore, be efficiently treated in rotor-dynamic transient anal· ysis.

Acknowledgments This work was supported by the U.S. Army Research Office, Con­

tract No. DAG 29-77-C-0009 and ERDA Contract No. EF-76-8-2479.

Journal of Lubrication Technology

References 1 Kirk, R. G., and MiUer, W. H., "The Influence of High-Pressure Oil Seal.a

on Turbo-Rotor Stability," ASLE Paper No. 77-LC-3A-l , ASLE/ASME Lu· brication Conference, Oct.. 1977.

2 Gunter, E. J ., Barrett, L. E., and Allaire, P . E., "Design of Nonlinear Squeeze-Film Dampers for Aircraft Engines," ASME JOURNAL OF LUBRI· CATION TECHNOLOGY, Vol. 99, No. l, Jan. 1977, pp. 57-64.

3 Lund, J. W., "'Rotor-Bearing Design Technology; Part Ill: Design Hand book for Fluid-Film-Type Bearings," Report No. AFAPL-TR-65-45; Part Ill, Wright-Patterson Air Force Base, 1965.

4 Allaire, P. E., Nicholas, J . C., and Gunter, E . J., "Systems of Finite El · ements for Finite Bearings," ASME JOURNAL OF LUBRICATION TECH NOL· CX::Y, Vol. 99, No. 2, Apr. 1977, pp. 187-197.

5 Hays, D. F., "A Variational Approach to Lubrication Problems and the Solution of the Finite Journal Bearing.'' ASME Journal of &uic Engineering, Vol. 81 , 1959, pp. 13-23.

6 Allaire, P . E., Barrett, L. E., and Gunter, E. J., " Variational Method for Finite Length Squeeze Film Damper Dynamics with Applications," Wear, Vol. 42, No. 1, 19'77, pp. 9-22.

7 Li, D. F., Allaire, P . E., and Barrett, L. E., "Analytical Dynamics of Partial Journal Bearings with Applications," ASLE Paper No. 77-LC-lA· l, ASLE/ASME Lubrication Conference, Oct. 1977.

8 Childs, D., Moes, H., and van Leeuwen, H., "Journal Bearing Impedance Descript.ions for Rotordynamic Applications," ASME JOURNAL OF LUBRI· CAT ION TECHNOLOGY, Vol. 99, No. 2, Apr. 1977, pp. 19S-214.

9 Booker, J . F., '"Dynamically Loaded Journal Bearings: Mobility Methods or Solution," ASME Journal of Basic Engineering, Sept.. 1965, pp. 537- 546.

10 Kirk, R. G., and Gunter, E. J ., "Short Bearing Analysis Applied to Rotor Dynamics; Part 1: Theory," ASMEJOURNAL OF LUBRICATION TEcHNOLOGY, Vol. 98, No. I, Jan. 1976, pp. 47- 56.

11 Kirk, R. G., and Gunter, E. J ., "Short Bearing Analysis Applied to Rotor Dynamics, Part 2: Results of Journal Bearing Response," ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 98. No. 2. Apr. 1976, pp. 319-329.

12 Falkenhagen, G. L., and Gunter, E . J., "Stability and Transient Motion of Vertical Three-Lobe Bearing Systems," ASMEJournal of Engineering for Industry, May 1972, pp. 665--677.

13 Massey, B. S ., "'A Note on the Theory or the Short Journal Bearing," International Journal of Mechanical Engineering Education, Vol. 5, No. l, 1977' pp. 95-96.

14 Barrett, L. E., and Allaire, P . E., '"A Further Note on the Theory of the Short Journal Bearing," International Journal of Mechanical Engineering Education, Vol. 6, No. 1, Jan. 1978.

15 Ocvirk, F. W., '"Short Bearing Approximation for Full Journal Bearing5,'' NACA TN 2808, 1952.

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